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A matrix   A   is totally positive (or non‐negative) of order   k  , denoted    T   P  k(or    T   N  k    ), if all minors of size    ⩽  k    are positive (or non‐negative). It is well known that such matrices are characterized by the variation diminishing property together with the sign non‐reversal property. We do away with the former, and show that   A   is    T   P  kif and only if every submatrix formed from at most   k   consecutive rows and columns has the sign non‐reversal property. In fact, this can be strengthened to only consider test vectors in    R  k    with alternating signs. We also show a similar characterization for all    T   N  kmatrices — more strongly, both of these characterizations use a single vector (with alternating signs) for each square submatrix. These characterizations are novel, and similar in spirit to the fundamental results characterizing    T  P    matrices by Gantmacher–Krein ( Compos. Math . 4 (1937) 445–476) and   P  ‐matrices by Gale–Nikaido ( Math. Ann . 159 (1965) 81–93).  As an application, we study the interval hull    I  (  A  ,  B  )    of two    m  ×  n    matrices    A  =  (   a   i  j    )    and    B  =  (   b   i  j    )   . This is the collection of    C  ∈   R   m  ×  nsuch that each    c   i  jis between    a   i  jand    b   i  j    . Using the sign non‐reversal property, we identify a two‐element subset of    I  (  A  ,  B  )    that detects the    T   P  kproperty for all of    I  (  A  ,  B  )    for arbitrary    k  ⩾  1   . In particular, this provides a test for total positivity (of any order), simultaneously for an entire class of rectangular matrices. In parallel, we also provide a finite set to test the total non‐negativity (of any order) of an interval hull    I  (  A  ,  B  )   .

Sign non‐reversal property for totally non‐negative and totally positive matrices, and testing total positivity of their interval hull